Question: Determine how many solutions exist for the system of equations. ${4x-y = 10}$ ${-4x+y = 1}$
Convert both equations to slope-intercept form: ${4x-y = 10}$ $4x{-4x} - y = 10{-4x}$ $-y = 10-4x$ $y = -10+4x$ ${y = 4x-10}$ ${-4x+y = 1}$ $-4x{+4x} + y = 1{+4x}$ $y = 1+4x$ ${y = 4x+1}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 4x-10}$ ${y = 4x+1}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.